Question

cos^2theta/1-tan theta+sin^3theta/sin theta-cos theta=1+sin theta. cos theta, prove that.

Answer 1

Cos2theta/ (1- tan theta) + Sin3theta/(sin theta - cos theta)

Cos2theta [1 - (sin theta/cos theta)] - Sin3theta/(Cos theta - Sin theta)

Cos2theta . Cos theta/(Cos theta - Sin theta) -  Sin3theta/(Cos theta - Sin theta)  

Cos3theta /(Cos theta - Sin theta) -  Sin3theta/(Cos theta - Sin theta)  

(Cos3theta -  Sin3theta)/(Cos theta - Sin theta)

{a3 - b3} = (a - b) (a2 + ab + b2)

on applying above formula in numerator, we get:

(Cos theta -  Sin theta)(Cos2theta + Sin2theta + Cos theta.Sin theta) /(Cos theta - Sin theta)

(Cos2theta + Sin2theta + Cos theta.Sin theta)

(1 + Cos theta.Sin theta) = R.H.S